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Chinese Physics B

Optical wavelet-fractional squeezing combinatorial transform

Cui-hong Lv, Ying Cai, Nan Jin, Nan Huang

Published Aug 17, 2021• 3 citations
Publisher

Abstract

By virtue of the method of integration within ordered product (IWOP) of operators we find the normally ordered form of the optical wavelet-fractional squeezing combinatorial transform (WFrST) operator. The way we successfully combine them to realize the integration transform kernel of WFrST is making full use of the completeness relation of Diracʼs ket–bra representation. The WFrST can play role in analyzing and recognizing quantum states, for instance, we apply this new transform to identify the vacuum state, the single-particle state, and their superposition state.

DOI
10.1088/1674-1056/ac1e1c

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By virtue of the method of integration within ordered product (IWOP) of operators we find the normally ordered form of the optical wavelet-fractional squeezing combinatorial transform (WFrST) operator. The way we successfully combine them to realize the integration transform kernel of WFrST is making full use of the completeness relation of Diracʼs ket–bra representation. The WFrST can play role in analyzing and recognizing quantum states, for instance, we apply this new transform to identify the vacuum state, the single-particle state, and their superposition state.
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By the newly developed technique of integration within an ordered product (IWOP) of operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We can not only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area to finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the IWOP technique are fully used. All these confirms Dirac's assertion: " ... for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory".